Universitat Pompeu Fabra, 1998/1999
Due October 21, 1998
1. Consider a model with three possible effort levels E={e1,e2,e3}. There are two possible outcomes: x=10 and y=0. The conditional probabilities of the outcomes given the effort are px(e1)=2/3, px(e2)=1/2, px(e3)=1/3. The cost function of the effort is: v(e1)=5/3, v(e2)=8/5, v(e3)=4/3. And u(w)=w1/2 and B(x-w)=x-w.
2. Consider the problem of moral hazard when the agent is risk averse and has mean-variance preferences, that is, E(UA)=E(w)-(1/2)Var(w)-(1/2)e2 and has a reservation utility level U=0. The principal is risk neutral. The firm's sales are proportional to the agent's effort: x=e×H; where the mean of the random variable H is E(H)=m>0 and the variance V(H)=s2. The principal offers a wage contract of the form w(x)=A+Bx. Recall that for any random variable z and non- random variables C,D, E(Cz+D)=CE(z)+D, V(Cz+D)=C2V(z)=C2E((z-E(z))2).
3. In some universities the grades of the students are determined by by relative system. For example, the top 10% receive the highest grade, the next 30% receive the second highest grade, and so on.